This paper is a sequel to [1]} in which the convergence of symmetric, continuous time stochastic differential games to mean field games was analyzed.
Here the goal is to complement the convergence results by deriving large deviation principles (in Laplace form) for the sequence of Nash equilibrium and the associated state processes.
Let us briefly describe the stochastic differential game we consider, in its non-cooperative version.
The cooperative version is discussed in the last section of the article.
Let \(d \in \mathbb {N}\)  be fixed and let \((\Omega ,\mathcal {F}, \mathbb {P})\)  be a probability space which is rich enough to carry a (fixed) sequence of \(d\) -dimensional independent Brownian motions \((W^i)_{i\ge 1}\) .
For every \(N \in \mathbb {N}\) , consider the filtration \(\mathcal {F}^N:=(\mathcal {F}_t^N)_{t\in [0,T]}\) , which is the completion of \(\sigma (W^i_s, s\le t, i=1,\dots , N)\) .
Without further mention, we will always use the identification
\(W \equiv W^1 \quad \text{and}\quad \mathcal {F}\equiv \mathcal {F}^1.\) 
